Theorem 20 (Co-area Formula [1]})  
Let \(\phi : \mathbb {R}^n \rightarrow \mathbb {R}^m\)  be a Lipschitz function with \(n \ge m\) . Then, for \(A \subset \mathbb {R}^m\) ,
\(\int _{\phi ^{-1}(A)} \sqrt{\det (D\phi (x) D \phi (x)^T)} d \mathcal {H}^m(x) = \int _{A} \operatorname{vol}_{n - m}(\phi ^{-1}(y)) d \mathcal {H}^m(y).\) 

